
I 



s& 



•^C*4*S^ 



NOTES AND EXERCISES 



ON 



SURVEYING 



• 

FOR THE USE OF STUDENTS IN 



KENYON COLLEGE 



BY 



ELI T. TAPPAN, LL. D. 

PROFESSOR OF MATHEMATICS. 






MT. VERNON, OHIO: 
Republican Steam Printing House, 

1881. 






Copyright, 1878, by ELI T. TAPPAN. 






S~3ff43 






SURVEYING 



Surveying is that branch of applied mathematics which has for its 
object the measurement of the earth's surface, including heights, dis- 
tances, and areas, and the representation of the same by a plat or 
chart. 

Plane surveying is that in which small portions of the earth's sur- 
face are regarded as parts of a plane. Topographical surveying con- 
sists in the measurement and exact delineation of a place, including 
variations of level. Geodetic surveying, or Geodesy, relates to the 
whole earth, or to large portions of it, taking into consideration the 
curvature of the surface. 

Surveying is to be distinguished, on the one hand, from mensura- 
tion, which is applied to the measurement of smaller objects, and on 
the other hand, from Astronomy, which relates to the sizes and dis- 
tances of the heavenly bodies. 

All measurement consists in two operations — observation and calcu- 
lation. The results of observation are more or less imperfect, depend- 
ing upon the keenness of the senses and manual skill. Calculation 
may be made to any required degree of accuracy. 

An observation in surveying consists in measuring either a distance 
or an angle. These quantities, of both kinds, are observed by the di- 
rect application of a standard. In geodesy, however, as in astronomy, 
in addition to distances and angles, time is an object of direct observa- 
tion. 



Note. — This is not a treatise on Surveying. Many things are omitted which 
can be taught better orally, either in the class-room or in the field. 



4 SURVEYING. 

STANDARDS OF LENGTH AND OF ANGLE. 

The standard measures of length are the meter and the yard. The 
meter is used in the geodetic surveys made by the government. The 
foot, divided decimally, is generally used by railway engineers. 

The meter, when first adopted by the French, was intended to be 
one ten-millionth part of the distance on the surface of the earth from 
the equator to the pole. More accurate geodetic surveys have shown 
that the meter in actual use is a little shorter than this. As the meter 
has been adopted by nearly all enlightened nations as the primary unit 
for all measures, it is not probable that any effort will be made to cor- 
rect the error. 

The standard meter is an actual bar of platinum kept by the French 
government. The standard yard is an actual bar kept by the British 
government. The government of the United States has very accurate 
copies of these, but it is said that the American yard exceeds the Brit- 
ish standard by nearly one-thousandth of an inch, and that there is a 
smaller error in the meter. 

In all instrumental observation, the standard of angular quantity is 
the degree. The grade (the hundredth part of a right angle) is only 
used in some calculations. 

INSTRUMENTS OF LENGTH. 

In the geodetic surveys in the United States, bars of six meters 
length are used. The contacts are regulated by screws, and thermom- 
eters are inserted in the bars for observation of the temperature of the 
metal. 

The common surveyors chain is four rods long, in one hundred 
links, but the instrument generally used is a half-chain. Railway en- 
gineers use a chain of one hundred feet. 

When the common surveyor's chain is used for measuring horizontal 
distance, the two ends must be held at the same level, whatever be the 
actual unevenness of the surface. Hence, it is necessary, in testing the 
chain, to allow for the diminution in length caused by the sagging 
when suspended. It also becomes the rule that all chaining is done by 
the chain held up. Evidently some skill is required to make uniform 
and correct measures of distance in this way. Sometimes, on a steep 
hill-side, it is necessary to use a smaller part of the chain. * 

In using the eleven pins with the chain, observe these rules: — 

i. One pin is put at the beginning of the distance to be measured. 



SURVEYING. 5 

2. The chainman in advance takes ten pins. 3. The pin in advance 
is fixed in its place before the pin at the rear is taken up, so that one 
pin is in place all the time. 

The following errors in chaining should be avoided: 

1. The rear chainman errs in not holding the end of the chain either 
against the pin or exactly over it. 

2. The forward chainman errs in not sticking the pin vertically 
under the end of the chain. If the chain is more than a foot from the 
ground, the pin must be dropped. 

3. Sticking a pin out of the line. The rear chainman should keep 
in view the signal at the end of the line; and should correct the posi- 
tion of the other chainman if he is right or left of the signal. 

4. Counting half-chains as chains. 

5. Counting a chain or half-chain too many. Let the last pin be- 
fore the end of the line remain in the ground till the distance is noted; 
then the number of pins in the rear chainman's hand shows the num- 
ber of lengths of the chain or half-chain. 

6. Writing the number of links, when less than ten, as tenths of a 
chain instead of hundredths. 

Skill in the use of these and all other instruments can be acquired 
only by practice, under the direction of a competent surveyor. 

Exercise. — Let a line be measured both by chain and by rod; let this be 
done by various members of the class. Compare the results, and consider the 
probable limits of error. 

The levelling staff, or rod, is used for observing differences of level, 
that is, vertical distance. In its use, it must be held as nearl} T vertical 
as possible. 

In the levelling instrument, the parts to be noticed are: — 

1. The tripod; 

2. The joint, and levelling screws; 

3. The vertical axis, revolving bar and wyes; 

4. The telescope and spirit level; 

5. The adjustment of the telescope parallel to the spirit level; 

6. The adjustment of the telescope perpendicular to the ver- 

tical axis. 

INSTRUMENTS OF DIRECTION. 

These are the compass, the transit, the theodolite, and the sextant. 
A horizontal angle is one having its arms in a horizontal plane. A 
vertical angle is one having its arms in a vertical plane, one of the 



6 SURVEYING. 

arms being horizontal. It is an angle of elevation or of depression* 
according as the other arm is above or below the horizon. 

The compass is used for measuring the horizontal angle which any 
line makes with the meridian. The engineer 's transit is used for meas- 
uring any horizontal angle. The theodolite is used for measuring both 
horizontal and vertical angles. The sextant is used for measuring an- 
gles in any plane whatever. 

In the compass, the parts to be noticed, are: i. The staff; 2. The 
joint; 3. The box; 4. The circle; 5. The needle; and 6. The sights- 
Some compasses have a spirit level. The solar compass has an appa- 
ratus for determining the true meridian by observations of the sun. 

In the transit there are to be noticed: — 

1. The tripod and the plummet; 

2. The joint and levelling screws; 

3. The vertical axis with its clamp; 

4. The graduated circle and vernier plate; 

5. The spirit level and the telescope. 

The theodolite has, besides the above, a vertical graduated circle and 
a vernier. 

The vernier is an ingenious contrivance for more accurate measure- 
ment. It is applied to the graduated standard of measure, to the grad" 
uated arc in angular measurement, and to the graduated bar or rod in 
linear measurement. 

The vernier moves along the principal scale, and is so divided that 
n divisions of it are equal to n — 1 or n-\-i of the smallest divisions of 
the scale. The zero point of the vernier indicates the position on the 
scale which is to be ascertained. If this exactly coincides with a di- 
viding mark, then the measure is read on the principal scale, and no 
reading of the vernier is required; but when the zero point of the ver- 
nier comes between two marks on the scale, the main scale is first read 
and there must be added an amount ascertained by the vernier. The 
amount to be added is indicated by that division mark on the vernier 
which exactly coincides with one on the scale. If this mark is the 
mth from zero, there must be added m times the nth part of the quan- 
tity indicated by one of the smallest divisions on the principal scale. 
This nth part is called the least count of the vernier. 

The explanation of the principle of the vernier is left to the student, 
an easy task when the instrument is in his hands. (In the theodolite 
of K. C, n is 30, and the least count is one minute of arc. In the ler- 
elling rod, n is 10, and the least count is one-thousandth of a foot.) 



SURVEYING. 7 

The following errors in observing angles should be avoided: 

(a) With the compass — i. Reading East for West, and vice versa; 
2. Reading the angle when the sights are not on the signal; and 3. 
Having the compass out of level. 

(b) With the theodolite — 1. Neglect to notice in which way the 
vernier plate is turned, and whether the zero mark is passed over; 

2. Not having the hair line on the signal when the angle is read; 

3. Not having the lower telescope* on its mark; and 4. Allowing the 
instrument to be out of level. 

(c.) With all instruments — Error of parallax in reading grades; i. e. 
holding the eye to one side. 

Exercises:— *• Make a drawing and a description of each instrument 
used by the class, specifying the use and purpose of the several parts. 

2. Measure with the theodolite, the three angles of a triangle. The difference 
between the sum and two right angles shows the amount of error. Consider this 
knowledge of total error and compare this work with that of measuring a line. 

3. Measure with chain and theodolite, two sides and the included angle of a 
iriangle; calculate the third side by trigonometry, and then measure it by chain. 
Consider in which of the measured elements was probably the greater source of 
error, taking into view all the circumstances. 

Students should be cautioned not to abuse instruments. Treat every 
instrument gently. Put no strain on a screw. If a grain of sand gets 
into a joint, let the instrument be taken apart and cleaned; do not use 
a dirty joint. A chain should not be tangled. No more force than is 
needed to lift the links should be used in untangling a chain. 

INSTRUMENTS FOR PLATTING. 

Every student should be provided with a scale, a protractor, and a 
pair of dividers. 

Exercises. — 1. Draw any triangle on paper; measure each angle with 
the protractor. The difference between the sum and two right angles shows 
the amount of error in measurement. 

2. Measure two sides; from these and the angles calculate the third side; then 
measure and compare. This exercise may be varied to suit every case of equality 
of triangles. 

3. Make plats of work done in the field. 

HEIGHTS AND DISTANCES. 

The solution of the principal problems of this class, by triangulation, 
is explained in the Trigonometry, Art. 875. 

*Many theodolites do not have the lower telescope. 



8 SURVEYING. 

In applying these methods in the field, the student should endeavor 
to measure every line in at least two independent ways. 

LEVELLING. 

When great accuracy is required, as in mining and making railways 
and canals, differences of level are measured by the levelling in- 
strument. 

By setting the instrument at nearly equal distances from the preced- 
ing and the following station, the observer may avoid the effect of a pos- 
sible maladjustment of the instrument. Some authors advise that this 
be done in order to avoid the error arising from the curvature of the 
earth's surface. A little observation and calculation will show that no 
ordinary instrument is sufficiently accurate to detect any error that 
c ould be attributed to this cause. At the distance of 200 feet the de- 
flection is less than one-thousandth of a foot. 

When the object of levelling is simply to ascertain the difference in 
level of two points, and no plat is to be made, then the notes need re- 
cord only the "back sights and fore sights" in two columns. Thus, to. 
find the height of Rosse porch above Ascension door-sill: 

Back Sights. Fore Sights. 

1st 3.496 feet. 2.351 feet. 

2d 9.620 0.705 

3d 10.721 4.388 

23.837 7.444 

7.444 

16.393 feet. 

When, however, a profile is to be made, as of a road or canal, the 
level at every 100 feet, or other certain distance,- should be recorded. 
The profile usually has a larger scale, ten-fold or more, for the heights 
than for the horizontal distances. 

In a topographical map, contour lines are made; i. e., lines showing 
where level planes would intersect the actual surface, these planes be- 
ing at regular intervals of height above some established base. The 
proper location of such lines is found by levelling, the details of the 
work depending upon the peculiarities of each locality. 

Exercises.— 1. Ascertain the height of a hill by levelling over two dif- 
ferent routes. Compare, and discuss the probable error. 

2. Measure then the same height by triangulation, by at least two sets of ob- 
servations and calculations. Compare as before. 

3. Determine the relative degrees of accuracy of the two methods. 



SURVEYING. 9 

AREA OF LAND. 

In determining the area of land, the inequalities of the surface are 
disregarded, the area to be measured is that of the horizontal plane 
within the boundaries. The length of a side is the horizontal distance 
between its extremities. The angles are horizontal. 

When a piece of land is in the shape of a square, a rectangle, a par- 
allelogram, a triangle, a trapezoid, or any regular geometrical figure, 
the method of measuring its area consists of applying the geometrical 
principle. 

When the shape is that of a polygon, that is, when all the sides are 
straight, the usual method of surveying is by a system of triangles and 
trapezoids. This method is described in the following pages. When 
one of the boundaries is an irregular curve, as the bank of a stream, 
the error may be diminished at will by substituting straight lines near- 
ly coincident with the curve. Sometimes it is more convenient to di- 
vide such a tract, surveying the more irregular part by itself. The ad- 
vantage of this is in excluding many small sides from the calculation 
of the whole area. 

The Ji eld notes of a survey are the record, made on the spot, of the 
bearing and distance of each side of the field. For example, the fol- 
lowing are the field notes of the survey of Jan's Lot, a six-sided field: 





Bearings. 


Distances. 


I. 


S. 8 9 ° E. 


5-335 


2. 


N. 26 E. 


i.6o 


3- 


N. 55° W. 


4-57 


4- 


S. 20° W. 


1.842 


5- 


S. 87° W. 


J-593 


6. 


S. 2° W. 


2.15 



The bearing ot a line is the angle, not over 90°, which the line makes- 
with the meridian, the letters indicating the quarter of the compass. 

The distance of a line or side of a field, is its length. 

The degree of accuracy required depends entirely upon circumstan- 
ces. Only very valuable land need be measured to a tenth of a link. 

The method of calculating the area is best explained by means of a 
plat. In practice, the plat may be made either before or after the cal- 
culation. 

First, make a line parallel to the side of the paper, for a prime me- 
ridian. This line should pass through either the easternmost or the 
westernmost corner of the plat. In the following plat, which illus- 
trates the given field notes, the meridian NS passes through the west- 
ern corner. 



10 



SURVEYING. 



N 



Scale: y 2 inch to the chain. 




With the protractor, make the angle BAS = 89 , that being 
the first bearing. Make the length of the side AB according to the 
scale. Since the direction of the next side is 26 East of North, and 
;the direction BA is 89 West of North, the angle ABC must be made 
1 15 . Since the next direction is 55 West of North, and CB is 26 
West of South, the angle BCD must be made 99 , that is 180 — (55°-f- 
26 ). So on, for the angles D, E, and F. The angle at each corner 
depends upon the preceding and the following bearing. It may be, 1. 
the sum of the bearings, or 2. the supplement of that sum, or 3. the 
difference of the bearings, or 4, the supplement of that difference. If 
the survey and drawing are accurate, the end of the last side falls 
upon the initial point A. 

When it is important to make an exact plate, it is better to draw a 
meridan line at every corner. The method given in the last para- 
graph renews at every angle the errors made at all the previous an- 
gles. A very accurate plat can be made, after calculating the lati- 
tudes and departures, by using them to lay off the sides of the field. 

From B, C, D, E, and F, let the perpendiculars BG, CH, DJ, EK, 
and FL, fall on the prime meridan NS. (These are not given in the 
diagram, but should be made by the student.) These perpendiculars 
form the bases of four trapezoids and two triangles. The other sides 
of these six figures are either sides of the field or parts of the prime 
meridian. From the observed bearings and distances, we may calcu- 



SURVEYING. 



11 



late the bases and altitudes, and thence the areas of all these figures; 
then substracting the sum of the four areas which are outside of the 
field from the sum of the other two, which contain these four and the 
field, we have the area of the field. See the calculation. 

Calculation of the Area. 



Latitudes. 



North. South 



1.438 
2.621 



4.059 



.093 



1.731 

.083 
2.149 



4.056 



Error 



Departures. 



East. 



5.334 
.701 



6.035 



West 



3744 
.636 

1.591 
.075 



6.046 



Error 



11 



Correct's 



+2 
— 1 



D. 



+4 

+1 
—3 
— 1 
— 1 
— 1 



Balanced. 



Lat. 



.095 
1.438 
2.620 
1.731 

.083 
2.149 



Dep's. 



+5.338 
+ .702 
—3.741 

— .635 
—1.590 

— .074 



D.M.D. 



11.378 

8.339 

3.'963 

1.738 

.074 



20) 



Double Areas 



North. South 



16.362 

21.848 



.507 



6.860 
.144 
.159 



38.210 

7.670 



30.54 



1.527 



7. 70 



The latitude of a side is the distance which one end is North or 
South of the other. Thus AG is the latitude of the side AB; HG 
is the latitude of BC, etc. The latitude of a side is its projection on a 
meridian line. It is the product of the distance by the cosine of the 
bearing. 

The departure of a line is the the distance which one end is East or 
West of the other. Thus GB is the departure of the side AB; HC — 
GB is the departure of BC, etc. The departure of a side is its projec 
tion on a parallel of latitude. Also, it is the product of the distance 
by the sine of the bearing. 

The latitudes and departures are ascertained by aid of a traverse 
table. This is a table of the latitudes and departures when the distance 
is unity, at every quarter of a degree of the quadrant. The traverse 
table used by professional surveyors gives the latitudes and departures 
with the multiples for the first nine digits. The one on pages 15 and 
16, is sufficient for learning the principle. 

The latitudes and departures are distinguished as Northings, South, 
ings, Eastings, or Westings, according to the bearings, and are placed 
in four separate columns. If the survey and calculations are accurate, 
the sum of the northings must'Tb'e equal to the southings, and the sum 
of the eastings to that of the westings. Equality in these is not to be 
expected, for the observations are not perfect, the angles being rarely 



12 SURVEYING. 

measured to less than one-fourth of a degree, or the sides to less than 
one link. Whether the error may be disregarded is a question of 
economy, which depends upon the amount of the error and the value of 
the land. If the error is not too great, it should be distributed, so as to 
make the northings equal to the southings, and the eastings equal to 
the westings. 

When there is no reason to suspect that the error belongs more to 
one side than to an other, it may be distributed in, proportion to the 
distance. For example, in the above survey, the sum of the distances 
is 17 chains and 9 links; the error of latitude 3, and the error of depart- 
ure 11, may be distributed as in the two columns of corrections. 
Notice that as the southings are less than the northings, a correction 
applied to a southing is marked -}-, and one applied to a northing — . 
The same principle is used in the signs of the corrections of 
departures. 

This mode of distributing the error is well enough under the cir- 
cumstances stated, but the surveyor who has been over the whole line 
with chain and compass usually knows that the error is probably due 
to the sides, in a different ratio from that of their lengths. On a short 
side there may be many obstructions and difficulties, none of which are 
met on a long side. The skilful surveyor distributes the error accord" 
ing to his judgment of all the circumstances. 

The figures in the columns of balanced latitudes and departures are 
found by applying the corrections to the previous columns. They are 
"''balanced" because the sum of the northings equals that of the south- 
ings, etc. 

The balanced departures of those sides which tend or depart from 
the prime meridian are marked +, and those which tend towards 

"it . When the prime meridian is on the east, the westings are posi- 

ive and the eastings are negative. 

The double meridian distance of a side is twice the distance of the 
middle point of that side from the prime meridian. 

In the case of the first side, the double meridian distance is equal to 
the departure, as appears by the plat. The same is true for the last 
side. For each of these two, the double meridian distance is the base 
of the triangle; for every other side, it is the sum of the bases of the 

trapezoid. ^uC& 

The double meridian distance of any 4^^. is found by the following 

"Rule —Add to the double meridan distance of the preceding side the sum 
of the departures of both. The algebraic sum is intended, that is, a negative de- 
parture must be substracted. 



SURVEYING. 13 

The proof of this rule is the geometric truth: the middle point of 
any side is farther (algebraically) from the prime meridian than the 
middle point of the preceding side, by half the sum of the departures 
of the two sides. Thus, the middle point of BC is farther from NS 
than the middle point of AB, by half the sum of the departures of AB 
and BC. Likewise, the middle point of CD, etc., but notice that the de- 
parture of CD is negative. 

If the departure of the last side is not equal to the double meridian 
distance found by this rule, there is some error in the work. 

Now we have the altitudes -and the bases of all the triangles and 
trapezoids — the altitudes being the same as the latitudes, and the bases 
or sum of the bases in each case being the double meridian distance. 

The product of the base, or the sum of the bases, by the altitude 
gives the double area. Subtracting the sum of the exterior area s 
from the sum of those which include the whole figure, (in this case the 
South areas from the North) the remainder is twice the area of the 
of the field, expressed in square chains. Dividing this by 20 reduces 
it to acres. 

It may be that the first station given in the field notes is not either 
the easternmost or the westernmost corner of the field. By an inspec- 
tion of the departures the surveyor sees which are these two corners, 
and he begins the calculation of the D.M.D. at one of them. If he be- 
gins at any other corner, at least one of the double meridian distances is 
negative, which produces a needless difficulty in the calculation. 

As a general rule, the latitudes and departures should be calculated 
to one more decimal place than is used in the field notes. 

Exercises,— !• Make a plat and calculate the area of the lot having this 
boundary : Beginning at a certain corner, thence South 8}4 degrees East 22 
chains and 81 links ; thence North 53 deg. West 23 chains and 21 links ; thence 
North 123^ deg., West 13 chains and 20 links; thence North 65 deg., East 17 
chains and 82 links ; thence South 93^ deg., East 12 chains to the place of the 
beginning. 

2. Divide this lot into halves by a line from the corner where the field notes 
begin. 

3. Run an East and West line through this lot, leaving two acres on the 
North side. 

4. The field notes of one side of a four-sided field are lost ; the remaining 

three are : 

North 24 West 34. chains 

North 50 East 28.34 " 

South 4 East 34.20 

Supposing these to be correct, what are the 4th bearing and distance ? 



14 



5. 







SURVEYING. 




)er 


of hectares within the following boundai 


i 


S. 


5S°i 


W. 


292.7 meters. 


2 


N. 


34° 


W. 


19S ,; 


3 


N. 


8i°i 


w. 


212.4 '• 


4 


N. 


36°i 


E. 


247.3 


5 


N. 


1 4 


E. 


115,8 


6 


N. 


79°i 


E. 


154. 


7 


S. 


86°| 


E. 


205.5 


8 


S. 


I2°i 


W. 


181.5 


9 


s. 


250 


E. 


219.2 " 



6. After surveying a field, it was discovered that the half-chain used was 33 
feet 4 inches long ; what correction was necessary to determine the true area? 

7. Calculate this: From beginning N. 10° W. 18 chains; thence N. 20° 
E. 20 chains; thence S. 89° E. 30 chains; thence S. 9° E. 36% chains ; thence W„ 
39 chains to place of beginning. 



TRAVERSE TABLE. 



Bea'g 


Latitude 


Depart' e 




Bea'g 

o 


Latitude 


Depart'e 


o 






o 






o 


1 


1.0000 


0.0044 


8 

4 


Ui 


0.9799 


0.1994 


1 
2 


1 
f 

1 


1.0000 
.9999 


.0087 
.0131 


1 
2 
1 
4 


3 
4 

12 


.9790 
.9781 


.2036 

.2079 


1 
4 

78 


1 


.9998 


.0175 


89 


i 

4~ 


.9772 


.2122 


3 
4 


1 
4 


.9998 


.0218 


3 
4 


l 
"2" 


.9763 


2164 


i 


1 
"2" 


.9997 


.0262 


1 
2 


3 
4 


.9753 


.2207 


1 
4 


1 


.9995 


.0305 


1 
4 


13 


.9744 


.2250 


77 


2 


.9994 


.0349 


88 


i 

4 


.9734 


.2292 


f 


i 

4 


.9992 


.0393 


3 
4 


1 
2 


.9724 


.2334 


i 


1 
a 


.9990 


.0436 


1 
2 


3 
4 


.9713 


.2377 


1 

4 


a 

3. 

A 


.9988 


.0480 


1 
4 


14 


.9703 


.2419 


76 


3 


.9986 


.0523 


87 


i 

4 


.9692 


.2462 


3 

4 


i 
4 


.9984 


.0567 


~4 


1 
2 


.9681 


.2504 


1 
2 


i 

"2^ 


.99X1 


.0610 


i 


3 
4 


.9670 


.2546 


1 

4 


3 

4 


.9979 


.0654 


i 

4 


15 


.9659 


.2588 


75 


4 


.9976 


.0698 


86 


i 

4 


.9648 


.2630 


8 

4 


i 

4 


.9973 


.0741 


"if 


1 
2 


.9636 


.2672 


1 
2 


1 


.9969 


.0785 


l 
2 


3 

4 


.9625 


.2714 


1 
4 


3 

4 


.9966 


.0828 


1 
4 


16 


.9613 


.2756 


74 


5 


.9962 


.0872 


85 


i 

4 


.9600 


.2798 


8 

4 


i 

4 


.9958 


.0915 


3 
4 


1 
2 


.9588 


.2840 


1 
2 


l 

2 


.9954 


.0958 


1 

9 


8 
4 


.9576 


.2882 


1 
4 


3 
4 


.9950 


.1002 


J 

4 


17 


.9563 


.2924 


73 


6 


.9945 


.1045 


84 


i 

4 


.9550 


.2965 


3 
4 


i 

4 


.9941 


.10S9 


8 

4 


1 
"2" 


.9537 


.3007 


1 
2 


1 

f 
3 

i 


.9936 
.9931 


.1132 
.1175 


1 
-?> 

1 
4 


3 

4 

18 


.9524 
.9511 


.3049 
.3090 


i 
4 

72 


7 


.9925 


.1219 


83 


i 

4 


.9497 


.3132 


8 

4 


i 

4 


.9920 


.12d2 


3 

4 


1 
"2" 


.9483 


.3173 


i 


1 
"2" 


.9914 


.1305 


1 


3 

4 


.9469 


.3214 


1 
4 


3. 
4 


.9909 


.1349 


1 

4 


19 


.9455 


.3256 


71 


8 


.9903 


.1392 


82 


i 

4 


.9441 


.3297 


3 
4 


i 

4 


9897 


.1435 


3 

4 


1 
•J 


.9426 


.3338 


I 


1 

f 
8 


.9890 

.9884 


.1478 
.1521 


1 
2 
1 
4 


20* 


.9412 
.9397 


.3379 
.3420 


1 
4 

70 


9 


.9877 


.1564 


81 


1 
4 


.9382 


.3461 


3 
4 


i 

4 


.9870 


.1607 


4 


1 


.9367 


.3502 


i 


1 
"2" 


.9863 


.1650 


1 

■7T 


Ji 
4 


.9351 


.3543 


1 

4 


ii 

4 


.9856 


.1694 


1 

4 


21 


.9336 


.3584 


69 


10 


.9848 


.1736 


80 


i 


.9320 


.3624 


8 

4 


I 

4 


.9840 


.1779 


3 
4 


i 


.9304 


.3665 


i 


1 


.9833 


.1822 


1 
"7 


i 


.9288 


.3706 


i 


i 


.9825 


.1865 


1 
4 


22 


.9272 


.3746 


68 


11 


.9816 


.1908 


79 


i 


.9255 


.3786 


4 


* 


.9808 


.1951 


4 


i 


.9239 


.3827 


i 




Depart'e 


Latitude 


Bea'g 


1 


Depart'e 


Latitude 


Bea'g 



TRAVERSE TABLE. 



Bea'g 


Latitude 


Depart 'e 




Bea'g 

o 

34 


Latitude 


Depart'*.- 




o 
4 


0.9222 


0.3867 


o 

l 
4" 


0.8290 


0.5592 


o 

56 


23 


.9205 


.3907 


67 


i 


.8266 


.5628 


A 
4 


i ; 


.9188 


.3947 


3 
4 


1 
2 


.8241 


.5664 


1 
2 


1 
o 


.9171 


.3987 


1 
2 


4 


.8216 


.5700 


i 


3 

/I 


.9153 


.4027 


1 
4 


35 


.8192 


.5736 


55 


24 


.9135 


.4067 


66 


i 

4. 


.8166 


.5771 


A 
4 


i 
4 


.9118 


.4107 


3 
4 


2 


.8141 


.5807 


1 
2 


i 

o 


.9100 


.4147 


1 
"2" 


3 
4 


.8116 


.5842 


1 
4 


A 


.9081 


.4187 


1 

4 


36 


.8090 


.5878 


54 


4 

25 


.9063 


.4226 


65 


i 

4 


.8064 


.5913 


3 
4 


i 
4 


.9045 


.4266 


.a 

4 


1 
2 


.8039 


.5948 


\ 


i 


.9026 


.4305 


1 
2 


A 
4 


.8013 


.5983 


JL 

4 


3 


.9007 


.4344 


1 
4 


37 


.7986 


.6018 


53 


26 


.8988 


.4384 


64 


i 

4 


.7960 


.6053 


3 
4 


i 


.8969 


.4423 


A 
4 


4 


.7934 


.6088 


i 




.8949 


.4462 


1 


3 
4 


.7907 


.6122 


i 


3 


.8930 


.4501 


1 
4 


38 


.7880 


.6157 


52 


27 


.8910 


.4540 


63 


i 

4 


.7853 


.6191 


3 
4 


i 

4 


.8890 


.4579 


A 
4 


1 
2 


.7826 


.6225 


1 

"2" 


1 


.8870 


.4617 


1 
"2" 


3 

4 


.7799 


.6259 


1 
4 


3 

4 


.8850 


.4656 


I 


39 


.7771 


.6293 


51 


28' 


.8829 


.4695 


62 


i 

4 


.7744 


.6327 


A 
4 


i 

4 


.8809 


.4733 


A 
4 


1 
2 


.7716 


.6361 


1 

2 


1 
2 


.8788 


.4772 


1 
■2" 


3 
4 


.7688 


.6394 


1 
Z 


3 

A 


.8767 


.4810 


i 


40 


.7660 


.6428 


50 


29 


.8746 


.4848 


61 


4 


.7632 


.6461 


1 


i 

4 


.8725 


.4886 


3 
4 


1 
2 


.7604 


.6494 


i 


J 


.8704 


.4924 


1 
2~ 


3 
4 


.7576 


.6528 


i 


3 


.8682 


.4962 


1 
4 


41 


.7547 


.6561 


49 


30 


.8660 


.5000 


60 


i 
5 


.7518 


.6593 


4 


i 

4 


.8638 


.5038 


A 
4 


i 

2 


.7490 


.6626 


1 
2 


1 

o 


,8616 


.5075 


4 


3 
4 


.7461 


.6659 


1 
4 


3. 


.8594 


.5113 


i 


42 


.7431 


.6691 


48 


4 

31 


.8572 


.5150 


59 


i 

4 


.7402 


.6724 


1 


i 

4 


.8549 


.5188 


A 
4 


1 
2 


.7373 


.6756 


i 


1 

~?7 


.8526 


.5225 


1 

2 


I 


.7343 


.6788 


i 


3. 


.8504 


.5262 


.1 

4 


43 


.7314 


.6820 


47 


4 

32 


.8480 


.5299 


58 


i 

4 


.7284 


.6852 


A 

4 


i 

A. 


.8457 


.5336 


3 

4 


I 


.7254 


.6884 


1 
2 


1 


.8434 


.5373 


J 


A 

4 


.7224 


.6915 


i 


2 

3. 


.8410 


.5410 


i 


44 


.7193 


.6947 


46 


4 

33 ■ 


.8387 


.5446 


57* 


i 

4 


.7163 


.6978 


3 
4 


4 


.8363 


.5483 


| 


4 


.7133 


.7009 


4 


4 
1 


.8339 


.5519 


i 


4 


.7102 


.7040 


.1 

4 


2 

3 
4 


.8315 


.5556 


Bea'g 


45 


.7071 


.7071 


45 




Depart'c 


, Latitude 


Depart' e 


Latitude 


Bea'g 



LIBRARY OF CONGRESS 



020 186 81 



29 




V 




